TITLE: Can class number decrease as we grow over some fixed number field?
QUESTION [10 upvotes]: Is it possible that a number field $K$ has non trivial class group, while one of its finite extension has trivial class group ? 
Or may be, is it known Hilbert class field of which number fields have trivial class group ?

REPLY [11 votes]: The OP first question is whether an algebraic number field $k$ with class number $h_k >1$ can be embedded in an algebraic number field $K$ with class number $h_K = 1$. Numerical examples exist (see (1) or the previous answers). The second question is related to the so called "class field tower" problem. The "principal divisor theorem" of CFT states that every ideal of $k$ becomes principal in the Hilbert class field $k_1$, but the question is, if we repeat the process, i.e. we consider the tower $k \subset k_1 \subset k_2 ...$, where $k_{i+1}$ is the Hillbert class field of $k_i$, is the extension $k_{\infty}=\cup k_i$ finite over $k$ ? If the above embedding problem with $h_K =1$ has a solution, then $k_{\infty} \subset K$ and the CF tower is finite. Conversely, if the CF tower is finite, then $k_{\infty}$ is the smallest $K$ with $h_K=1$ in the problem above.
Examples of finite CF towers exist, and it was even believed that all CF towers are finite (see (1)). But in the $60$'s, Golod and Shafarevitch constructed examples of infinite CF towers using cohomological methods. The G-S. theorem rests on the following criterion: There exists a function $\gamma(n)$, e.g. $2++2\sqrt {n+1}$, such that the dimension over $\mathbf F_p$ of $Cl_k /p$ is < $\gamma(n)$ for any algebraic number field $k$ of degree $n$ whose $p$-CF tower (i.e. only $p$-extensions are considered in the CF tower) is finite (see (2), chapter 9). Since then, many refinements have been proved, but the approach remains fundamentally the same.
(1) F. Lemmermeyer, Class Field Towers, 2010
(2) Cassels-Fhröhlich, Algebraic Number Theory, Acad. Press, 1967